4.2五大運算基本微分公式

 

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4.1 微分定義
4.2五大運算基本微分公式
4.3指數函數之微分
4.4對數函數之微分
4.5對數微分法

 

 

4-2 五大運算基本微分公式

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一、常數法則

常數函數之導數為零

 

二、冪次方法則(Power rule)

多項式微分公式推導

已知                               

代入導函數定義式       

                                   

二項式定理展開           

       

整理得                             

最後得                             

所以                

 

三、常數乘法法則

是可微函數,且為常數,則

 

四、加法與減法法則

若已知為可微分函數,則下列運算式仍成立:

函數和微分公式           

函數差微分公式   

1.      Given a function ,.    calculate

解答:

 

 

2.      ,則 

 

解答:

微分                           

                               

                               

 

3.    ,求=?

 

解答:

已知             

微分             

代入             

 

 

 

五、乘法法則

若已知為可微分函數

函數積微分公式(Product Rule       

 

代入導函數定義得       

 

六、除法法則

函數商微分公式(Quotient Rule     

亦表示:                

 

4.    ,則?

 

解答:

已知函數積之微分  

已知             

代入             

               

 

 

 

 

 

5.    ,則?

 

解答:

函數商微分公式(Quotient Rule     

 

 

 

 

七、連鎖律( Chain Rule )

的可微分函數,且的可微分函數,

則合成函數的可微分函數且

也可寫成 

 

6.    ,則?

 

解答:

 

 

7.      Find the first and second derivatives of the functions.

   a.

   b.

解答:

     a.

      

         

           .

      

               

               

                 .

     b.

      

        .

        .

 

8.      Show that  .

 

解答:

     If  ,

     If  ,

     Therefore, we can show that  for all .

 

 

9.      Find all values of  for which the tangent lines to the graphs of  and  at  and  are parallel.

 

解答:

     The slop of tangent line for  at  is .

     The slop of tangent line for  at  is  .

    

      

 

 

10.      Find the value of  at the given value of .

     

解答:

    

    

     Therefore,

     .

 

 

11.  An oil slick spreads so that it forms a disk centered about the point of contamination. As the fixed volume of spilled oil disperses and the thickness of the slick decreases, the radius of the slick increases at the rate of 2 ft/min. How fast is the area of polluted surface water growing when the radius is 500 ft ?

 

解答:

     The area of polluted surface water is ,  is the radius.

     The radius  of the slick is a function of time , and its rate is given by

                     ft/min.

     Therefore, the rate of dispersing of the area of polluted surface water is

                    .

     Substituting radius  ft into  gives

                     ft2/min.

     Therefore, the area of polluted surface is increasing at the rate of  ft2/min 

     when radius is 500 ft.

 

 

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